On a nonlocal Cahn–Hilliard model permitting sharp interfaces

Burkovska, Olena; Gunzburger, Max D.

doi:10.1142/s021820252150038x

On a nonlocal Cahn–Hilliard model permitting sharp interfaces

Journal Article · Mon Jul 05 00:00:00 EDT 2021 · Mathematical Models and Methods in Applied Sciences

DOI:https://doi.org/10.1142/s021820252150038x· OSTI ID:1831700

Burkovska, Olena ^[1]; Gunzburger, Max D. ^[2]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Florida State Univ., Tallahassee, FL (United States); Univ. of Texas, Austin, TX (United States)

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. In this work, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

View Accepted Manuscript (DOE)

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC05-00OR22725; SC0021077

OSTI ID:: 1831700

Journal Information:: Mathematical Models and Methods in Applied Sciences, Journal Name: Mathematical Models and Methods in Applied Sciences Journal Issue: 09 Vol. 31; ISSN 0218-2025

Publisher:: World ScientificCopyright Statement

Country of Publication:: United States

Language:: English

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